Question

Find the least number which when divided by 35 gives a remainder 25, when divided by 45 leaves a remainder 35, and when divided by 55 leaves a remainder 45.
A. 3465
B. 4575
C. 3455
D. 3670

Answer 1

Hint: We will use Euclid’s division lemma which says that if two positive integers a and b, then there exists unique integers q and r such that which satisfies the condition
$a = bq + r$ where r is a positive integer less than b. Here, r is the remainder, and q is the quotient.

Complete step-by-step answer:
Let the least number be N
We have to find a number such that when divided by 35 gives a remainder 25, when divided by 45 leaves a remainder 35, and when divided by 55 leaves a remainder 45. So, we will apply division lemma in all the three cases as-
$N = 35x + 25$
$N = 45y + 35$
$N = 55z + 45$
Where x, y and z are quotients with 35, 45 and 55 respectively. Now, we will add 10 to all the three equations as-
$N + 10 = 35x + 25 + 10 = 35x + 35 = 35(x + 1)...(1)$
$N + 10 = 45y + 35 + 10 = 45y + 45 = 45(y + 1)...(2)$
$N + 10 = 55z + 45 + 10 = 55z + 55 = 55(z + 1)...(3)$
From equations (1), (2) and (3) we can see that the number $(N + 10)$ can be written as the multiple of 35, 45 and 55 respectively. This means that the number $(N + 10)$ is the LCM of 35, 45 and 55.
So, we will find the LCM of 35, 45 and 55 as-
$\begin{align}
  &3\left| {\underline {35,\;45,\;55} } \right. \\
  &3\left| {\underline {35,\;15,\;55} } \right. \\
  &5\left| {\underline {35,\;5,\;55} } \right. \\
  &7\left| {\underline {7,\;1,\;11} } \right. \\
  &11\left| {\underline {1,1,11} } \right. \\
  &\;1\left| {\underline {1,1,1} } \right. \\
\end{align} $

So, the LCM of the three numbers is-
$LCM\left( {35,45,55} \right) = 3 \times 3 \times 5 \times 7 \times 11 = 3465$
But this is not the final answer. We assumed that-
$N + 10 = 3465$
$N = 3455$

This is the required answer. The correct option is C.

Note: The main mistake is that the students just find the LCM and mark it as the correct answer. We need to keep in mind that N is the least number which we assumed. But the LCM is equal to $N + 10$. Also, we should keep in mind that during prime factorization, we should only use prime factors such as 2, 3, 5 and so on. Using composite factors will result in the wrong answer.